Integrand size = 26, antiderivative size = 580 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\frac {(8 A b+a B) (e x)^{5/2} \sqrt {a+b x^3}}{4 a e^4}+\frac {3 \left (1+\sqrt {3}\right ) (8 A b+a B) \sqrt {e x} \sqrt {a+b x^3}}{8 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}-\frac {3 \sqrt [4]{3} \sqrt [3]{a} (8 A b+a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{a} (8 A b+a B) \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{16 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
-2*A*(b*x^3+a)^(3/2)/a/e/(e*x)^(1/2)+1/4*(8*A*b+B*a)*(e*x)^(5/2)*(b*x^3+a) ^(1/2)/a/e^4+3/8*(8*A*b+B*a)*(1+3^(1/2))*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b^(2/ 3)/e^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))-3/8*3^(1/4)*a^(1/3)*(8*A*b+B*a)*(a^ (1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+ 3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3 ^(1/2)))*EllipticE((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x *(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(e*x)^(1/2)*((a^(2/3)-a^(1 /3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/b^(2/3 )/e^2/(b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1 +3^(1/2)))^2)^(1/2)-1/16*3^(3/4)*a^(1/3)*(8*A*b+B*a)*(a^(1/3)+b^(1/3)*x)*( (a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2) /(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*EllipticF ((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^( 1/2),1/4*6^(1/2)+1/4*2^(1/2))*(1-3^(1/2))*(e*x)^(1/2)*((a^(2/3)-a^(1/3)*b^ (1/3)*x+b^(2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/b^(2/3)/e^2/ (b*x^3+a)^(1/2)/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/ 2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=-\frac {2 A x \left (a+b x^3\right )^{3/2}}{a (e x)^{3/2}}-\frac {4 \left (-4 A b-\frac {a B}{2}\right ) x^4 \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {b x^3}{a}\right )}{5 a (e x)^{3/2} \sqrt {1+\frac {b x^3}{a}}} \]
(-2*A*x*(a + b*x^3)^(3/2))/(a*(e*x)^(3/2)) - (4*(-4*A*b - (a*B)/2)*x^4*Sqr t[a + b*x^3]*Hypergeometric2F1[-1/2, 5/6, 11/6, -((b*x^3)/a)])/(5*a*(e*x)^ (3/2)*Sqrt[1 + (b*x^3)/a])
Time = 0.69 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {955, 811, 851, 837, 25, 766, 2420}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {(a B+8 A b) \int (e x)^{3/2} \sqrt {b x^3+a}dx}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3}{8} a \int \frac {(e x)^{3/2}}{\sqrt {b x^3+a}}dx+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3 a \int \frac {e^2 x^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3 a \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} e^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 e^2+\left (1-\sqrt {3}\right ) a^{2/3} e^2}{\sqrt {b x^3+a}}d\sqrt {e x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {(a B+8 A b) \left (\frac {3 a \left (\frac {\frac {\left (1+\sqrt {3}\right ) e^3 \sqrt {e x} \sqrt {a+b x^3}}{\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x}-\frac {\sqrt [4]{3} \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e \sqrt {e x} \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right ) \sqrt {\frac {a^{2/3} e^2-\sqrt [3]{a} \sqrt [3]{b} e^2 x+b^{2/3} e^2 x^2}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x e+\sqrt [3]{a} e}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} e x \left (\sqrt [3]{a} e+\sqrt [3]{b} e x\right )}{\left (\sqrt [3]{a} e+\left (1+\sqrt {3}\right ) \sqrt [3]{b} e x\right )^2}}}\right )}{4 e}+\frac {(e x)^{5/2} \sqrt {a+b x^3}}{4 e}\right )}{a e^3}-\frac {2 A \left (a+b x^3\right )^{3/2}}{a e \sqrt {e x}}\) |
(-2*A*(a + b*x^3)^(3/2))/(a*e*Sqrt[e*x]) + ((8*A*b + a*B)*(((e*x)^(5/2)*Sq rt[a + b*x^3])/(4*e) + (3*a*((((1 + Sqrt[3])*e^3*Sqrt[e*x]*Sqrt[a + b*x^3] )/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x) - (3^(1/4)*a^(1/3)*e*Sqrt[e*x]*( a^(1/3)*e + b^(1/3)*e*x)*Sqrt[(a^(2/3)*e^2 - a^(1/3)*b^(1/3)*e^2*x + b^(2/ 3)*e^2*x^2)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*EllipticE[ArcCos[(a ^(1/3)*e + (1 - Sqrt[3])*b^(1/3)*e*x)/(a^(1/3)*e + (1 + Sqrt[3])*b^(1/3)*e *x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*e*x*(a^(1/3)*e + b^(1/3)*e*x))/(a^( 1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)^2]*Sqrt[a + b*x^3]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*e*Sqrt[e*x]*(a^(1/3)*e + b^(1/3)*e*x)*Sqrt[(a^(2/3)*e^ 2 - a^(1/3)*b^(1/3)*e^2*x + b^(2/3)*e^2*x^2)/(a^(1/3)*e + (1 + Sqrt[3])*b^ (1/3)*e*x)^2]*EllipticF[ArcCos[(a^(1/3)*e + (1 - Sqrt[3])*b^(1/3)*e*x)/(a^ (1/3)*e + (1 + Sqrt[3])*b^(1/3)*e*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3 )*Sqrt[(b^(1/3)*e*x*(a^(1/3)*e + b^(1/3)*e*x))/(a^(1/3)*e + (1 + Sqrt[3])* b^(1/3)*e*x)^2]*Sqrt[a + b*x^3])))/(4*e)))/(a*e^3)
3.6.21.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Result contains complex when optimal does not.
Time = 4.83 (sec) , antiderivative size = 1123, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1123\) |
| elliptic | \(\text {Expression too large to display}\) | \(1161\) |
| default | \(\text {Expression too large to display}\) | \(5736\) |
-1/4*(b*x^3+a)^(1/2)*(-B*x^3+8*A)/e/(e*x)^(1/2)+(3*A*b+3/8*B*a)*(x*(x+1/2/ b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1 /2*I*3^(1/2)/b*(-a*b^2)^(1/3))+(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b ^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^ (1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b^2)^(1/3) +1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(- a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*( -a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)+1/2*I *3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(((-1/2/b*(-a*b^2 )^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/b^2*(-a*b^2)^(2 /3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*b/(-a*b^2)^(1/ 3)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1 /2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)) )^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a* b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^( 1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3 )))^(1/2))+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE (((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2 )^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),(...
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
Result contains complex when optimal does not.
Time = 3.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{6}\right )} + \frac {B \sqrt {a} x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {11}{6}\right )} \]
A*sqrt(a)*gamma(-1/6)*hyper((-1/2, -1/6), (5/6,), b*x**3*exp_polar(I*pi)/a )/(3*e**(3/2)*sqrt(x)*gamma(5/6)) + B*sqrt(a)*x**(5/2)*gamma(5/6)*hyper((- 1/2, 5/6), (11/6,), b*x**3*exp_polar(I*pi)/a)/(3*e**(3/2)*gamma(11/6))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{{\left (e\,x\right )}^{3/2}} \,d x \]